Biomedical Engineering Reference
In-Depth Information
The low-frequency approximation of a signal
f
at the scale
s
is computed as:
Lf
(
u
,
s
)
=
f
(
t
)
,φ
s
(
t
−
u
)
(6.9)
with
t
s
1
√
s
φ
φ
s
(
t
)
=
(6.10)
.
For a one-dimensional signal
f
, the continuous wavelet transform (6.7) is a two-
dimensional representation. This indicates the existence of redundancy that
can be reduced and even removed by subsampling the scale parameter
s
and
translation parameter
u
.
An orthogonal (nonredundant) wavelet transform can be constructed con-
straining the dilation parameter to be discretized on an exponential sampling
with fixed dilation steps and the translation parameter by integer multiples of a
dilation-dependent step [15]. In practice, it is convenient to follow a dyadic scale
sampling where
s
=
2
i
·
k
, with
i
and
k
being integers. With dyadic
dilation and scaling, the wavelet basis function, defined as:
and
u
=
2
i
t
−
2
j
n
2
j
1
√
2
j
ψ
ψ
j
,
n
(
t
)
=
(
j
,
n
)
∈
Z
2
,
forms an orthogonal basis of
L
2
(
R
)
.
For practical purpose, when using orthogonal basis functions, the wavelet
transform defined in Eq. (6.7) is only computed for a finite number of scales
(2
J
) with
{
J
=
0
,...,
N
}
, and a low-frequency component
Lf
(
u
,
2
J
) (often re-
ferred to as the DC component) is added to the set of projection coefficients
corresponding to scales larger than 2
J
for a complete signal representation.
In medical image processing applications, we usually deal with discrete data.
We will therefore focus the rest of our discussion on discrete wavelet transform
rather than continuous ones.
6.2.2 Discrete Wavelet Transform and Filter Bank
Given a 1D signal of length
N
,
{
f
(
n
)
,
n
=
0
,...,
N
−
1
}
, the discrete orthog-
onal wavelet transform can be organized as a sequence of discrete functions
according to the scale parameter
s
=
2
j
:
L
J
f
,
{
W
j
f
}
j
∈
[
I
,
J
]
,
(6.11)
where
L
J
f
=
Lf
(2
J
n
,
2
J
) and
W
j
f
=
Wf
(2
j
n
,
2
j
)
.
